# Question 16078: A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim [Return to catalogue page 81](https://ocme.evemisslab.com/en/questions/page/81/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations/#q-20261014-advance-20261101-r04-almost-sure-convergence-0053) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0031-16078 · q-20261014-advance-20261101-r04-almost-sure-convergence-0053 ## Original question and givens Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim). - G001 — Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-almost-sure-convergence](https://ocme.evemisslab.com/en/mko/mko-almost-sure-convergence/) ## Step-by-step derivation S001 · calculationBefore almost-sure full probability mass := sum(1/2, 1/2, 0) After almost-sure full probability mass = 1 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore same complete finite atomic space;raw masses=[[1, 2], [1, 2], [0, 1]];raw settling declarations=[3, 4, null] After normalized nonnegative masses=[1/2, 1/2, 0];total=1;integer N means equality to common limit at EVERY later index;null means model-declared nonconvergence Conditions: none Fix the same probability space and aligned atoms.An integer N declares equality on the entire later tail;null is model-declared nonconvergence,not an unchecked or missing receipt.These universal declarations cannot be manufactured from a finite observation. Step identifiers and dependenciesRule: asconv_declared_full_trajectory_model_not_missing_receipt · Uses: G001, S001 S003 · impliesBefore original atom=0;exact mass=1/2;full-tail declaration=3 After atom row=[0, 1/2, 3, Yes, Yes] Conditions: none Every positive-mass atom must have its convergent-tail declaration.A zero-mass failure may be ignored only for an almost-sure claim.Keep every atom's original order,mass and declaration;do not delete null atoms and pretend the statement holds everywhere. Step identifiers and dependenciesRule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S002 S004 · impliesBefore original atom=1;exact mass=1/2;full-tail declaration=4 After atom row=[1, 1/2, 4, Yes, Yes] Conditions: none Every positive-mass atom must have its convergent-tail declaration.A zero-mass failure may be ignored only for an almost-sure claim.Keep every atom's original order,mass and declaration;do not delete null atoms and pretend the statement holds everywhere. Step identifiers and dependenciesRule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S003 S005 · impliesBefore original atom=2;exact mass=0;full-tail declaration=null After atom row=[2, 0, null, No, Yes] Conditions: none Every positive-mass atom must have its convergent-tail declaration.A zero-mass failure may be ignored only for an almost-sure claim.Keep every atom's original order,mass and declaration;do not delete null atoms and pretend the statement holds everywhere. Step identifiers and dependenciesRule: asconv_each_positive_atom_obligation_and_null_exception · Uses: G001, S004 S006 · calculationBefore almost-sure declared nonconvergent exceptional mass := sum(0) After almost-sure declared nonconvergent exceptional mass = 0 Conditions: none Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore all model-declared nonconvergent atom indices=[2];their total mass=0 After positive-mass failures=[];almost-sure qualification=Yes Conditions: none On this complete finite atomic space,the bad set has mass0 exactly when it contains no positive-mass atom.That gives probability1 without changing actual nonconvergence on a null atom. Step identifiers and dependenciesRule: asconv_probability_one_not_every_outcome · Uses: G001, S006 S008 · impliesBefore selected original atom=2;verified mass=0;retained settling declaration=null After ignored only for almost-sure test=Yes;all positive atoms still audited;almost-sure qualification=Yes Conditions: none Verify actual zero mass before excluding this atom's effect on the almost-sure test.Its declaration remains visible.The ignored flag repairs neither pointwise failure nor failures at other positive atoms. Step identifiers and dependenciesRule: asconv_selected_actual_zero_mass_not_pointwise_repair · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute almost-sure declared nonconvergent exceptional mass with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Recompute almost-sure full probability mass with exact rational arithmetic, not rounded decimals. ## Result (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: e7d367dbb4323b8d19c48e2db37b0ee865ffb31e4b01fd4acceea21df8edd88b - candidate record SHA-256: 0374ebdf4d8352bb3dddc532105e83c123681d553eb4b5e8c13fb2a0f6e04e94 - semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a - question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0053.json)